260 XIV INTEGRATION IN LOCALLY COMPACT GROUPS index i. If /i is Lebesgue measure on R", and if jq = (at t,..., awl-), then we have (14.3.10.1) ju(P) = |det(av)|. For let u be the automorphism of the vector space R" given by u(et) = xt for 1 <£ z <£ « (where ^,..., £„ are the vectors of the canonical basis of Rn); we have P = w(K), where K is the parallelotope on the vectors el9 ...,£„. Hence from (14.3.6.2) and (14.3.9) we have ju(P) = |det u\ ju(K); but it follows imme- diately from (13.21.15) that /j,(K) = 1. Hence the formula (14.3.10.1). Now let S be the "simplex" on the vectors xl9,.., xn, i.e., the set of all n x = £ £i*i such that ti ^ ° for a11 *> and 5i + ' * * + €n ^ (14.3.10.2) ... BI Here again, by using the same linear transformation w, we are reduced to proving the formula (14.3.10.2) when xt = ei for 1 <* i ^ n. Let Sn denote the corresponding simplex, write \in in place of ju, and let an = t*n($n)* Identifying Rn with R""1 x R, consider for each AeR the section Sn(A) in R"""1. This set is empty if A < 0 or A > 1. If 0 :g A ^ 1, then Sn(A) is the set of points (£1, • • • > <^«-i) e R""1 such that and is therefore the image of Sn(0) under the homothety with ratio 1 — L Applying (14.3.6.2) and (14.3.9) to this homothety in R""1, we obtain Apply (13.21.8) to the compact set Sn, and we get Jo n Since clearly a^ = 1, we have an = l/nl. (14.3.11) Application to the calculation of integrals: II. Closed ball. With the same notation as in (14.3.10), we shall now calculate the measure ¥„ = nn(Bn) of the Euclidean unit ball Bn, i.e., the set of all (jct, ..., *„) e Rn Application to the calculation of integrals: I. Parallelotope and